| Description |
This submission contains VERT2LCON and LCON2VERT, which will find the linear constraints defining a bounded polyhedron in R^n, given its vertices, or vice versa. They are extensions of Michael Kleder's VERT2CON and CON2VERT functions that can handle cases where the polyhedron is not solid in R^n, i.e., where the polyhedron is defined by both equality and inequality constraints. Some other refinements to the original VERT2CON and CON2VERT subroutines, meant to improve speed and reliability, have also been made.
SYNTAX:
[A,b,Aeq,beq]=VERT2LCON(V,TOL)
[V,nr,nre]=LCON2VERT(A,b,Aeq,beq,TOL)
The rows of the N x n matrix V are a series of N vertices of the polyhedron
in R^n, defined by the linear constraints
A*x <= b
Aeq*x = beq
For LCON2VERT, Aeq=beq=[] are the default inputs, implying no equality constraints. The output "nr" lists non-redundant inequality constraints, and "nre" lists non-redundant equality constraints. Note further that A,b should be used to specify a solid region, while Aeq and beq should be used to further constrain the region to a sub-manifold of R^n, if needed. You should not use the inequality constraints to express a region in R^n that can be described by equality constraints.
The optional TOL argument is a tolerance used for rank-estimation purposes and to handle finite precision issues. Default=1e-10.
EXAMPLE:
Consider V=eye(3) corresponding to the 3D region defined by x+y+z=1, x>=0, y>=0, z>=0.
>> [A,b,Aeq,beq]=vert2lcon(eye(3))
A =
0.4082 -0.8165 0.4082
0.4082 0.4082 -0.8165
-0.8165 0.4082 0.4082
b =
0.4082
0.4082
0.4082
Aeq =
0.5774 0.5774 0.5774
beq =
0.5774
And the reverse conversion is,
>> V=lcon2vert(A,b,Aeq,beq)
V =
1.0000 0.0000 0.0000
-0.0000 1.0000 0
0 0.0000 1.0000 |
